On a generalization of the Brocard--Ramanujan Diophantine equation

Abstract

Let $Q_1,...,Q_r\in \mathbb{Z}[x]$ be polynomials having $0$ as a root. Let $f(x,y)\in\mathbb{Z}[x,y]$ be a homogeneous polynomial with factorization $f(x,y)=f_1(x,y)^{e_1}\cdots f_u(x,y)^{e_u}$, where $f_i(x,y)$ are irreducible homogeneous polynomials of degree $d_i\geq 2$. Fix some positive integers $A_1,...,A_r$. We show that under certain conditions, the diophantine equation $\prod_{i=1}^rQ_i(A_i^{n_i}n_i!)=f(x,y)$ has finitely many integer solutions.